A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

Random curves on surfaces induced from the Laplacian determinant

We define natural probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric. These measures arise from the Laplacian determinant and depend on the choice of a unitary connection on the tangent bundle to the surface. We show that, for a sequence of graphs $(G_n)$ conformally approximating the surface, the measures on CRSFs of $G_n$ converge and give a limiting probability measure on finite multicurves (finite collections of pairwise disjoint simple closed curves) on the surface, independent of the approximating sequence. Wilson's algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence. We set the framework for the study of these probability measures and their scaling limits and state some of their properties

Forman-Ricci flow for change detection in large dynamic data sets

We present a viable solution to the challenging question of change detection in complex networks inferred from large dynamic data sets. Building on Forman's discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci flow that already proved to be a powerful tool in image processing and graphics, to the case of undirected and weighted networks. The application of the proposed method on peer-to-peer networks yields insights into topological properties and the structure of their underlying data.Comment: Conference paper, accepted at ICICS 2016. (Updated version

Parabolic theory of the discrete p-Laplace operator

We study the discrete version of the $p$-Laplacian. Based on its variational properties we discuss some features of the associated parabolic problem. Our approach allows us in turn to obtain interesting information about positivity and comparison principles as well as compatibility with the symmetries of the graph. We conclude briefly discussing the variational properties of a handful of nonlinear generalized Laplacians appearing in different parabolic equations.Comment: 35 pages several corrections and enhancements in comparison to the v

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics